Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C5 · Schur multiplier (H2) = C24×C42 |
| direct product | C5×#13832 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C5 | C24 |
| commutator | C22 | C22×C42×C5 |
| Frattini, cosocle | C24 | C24×C5 |
| prime radical | C23×C5×D4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C5 ◁ G | |
1998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(6,7,9,8)(10,11,13,12)(15,18)(16,20)(22,23,25,26)(24,27,28,29);(10,11,13,12)(15,18)(19,21)(23,26)(24,28);(6,8,9,7)(10,12,13,11)(14,15)(16,19)(17,18)(20,21)(22,24)(23,27)(25,28)(26,29);(14,16)(15,19)(17,20)(18,21)(22,23)(24,27)(25,26)(28,29)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6820);